Modulational instability and position controllable discrete rogue waves with interaction phenomena in the semi-discrete complex coupled dispersionless system

Abstract

The position controllable discrete analytical higher-order rogue wave and periodic wave solutions in the semi-discrete complex coupled dispersionless system, as well as their mixed interaction phenomena, are theoretically provided in this study. Our first priority is to study the modulation instability theory study for this system and deduce the formation mechanism from its plane wave solutions. Then, we use the generalized (n,N−n)-fold Darboux transformation to create three types of position-controlled localized wave solutions, including rogue waves, periodic waves and their mixed interaction solutions. In particular, we find three different kinds of rogue wave structures in the same discrete system, including dark–bright structure, bright–bright structure, and the structure with two peaks and two depressions, which is a rare phenomenon in the field of nonlinear waves. All these innovative structures are discussed analytically and shown graphically. Besides, the principal structures and locations of these novel solutions can be controlled by a set of unique parameters so that we can theoretically manage them to appear anywhere on the plane wave by changing these parameters. These results may play a potential role to describe the interaction between the current-fed string and the external magnetic field and other physical phenomena.